The symmetric parabolic resonance

The parabolic resonance instability emerges in diverse applications ranging from optical systems to simple mechanical ones. It appears persistently in p-parameter families of near-integrable Hamiltonian systems with n degrees of freedom provided n + p ≧ 3. Here we study the simplest (n = 2, p = 1) symmetric case. The structure and the phase-space volume of the corresponding instability zones are characterized. It is shown that the symmetric case has six distinct non-degenerate normal forms, and two degenerate ones. In the regular cases, the instability zone has the usual O(ε) extent in the action direction. However, the phase-space volume of this zone is found to be polynomial in the perturbation parameter ε (and not exponentially small as in the elliptic resonance case). Finally, the extent of the instability zone in some of the degenerate cases is explored. Three applications in which the symmetric parabolic resonance arises are presented and analysed.